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Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

Is the mean of the squares of two numbers greater than, or less
than, the square of their means?

Tet-trouble

Stage: 4 Challenge Level:

Why do this problem?

This
problem offers an excellent opportunity for students to
practise visualisation and apply an idea normally only used in 2D
geometry to a 3D case. Learners will have to consider carefully how
to communicate their methods for testing combinations and that they
have considered all possibilities.

Possible approach

In silence, write three lengths on the board (for example 3
units, 6 units, 7units) and accurately draw a triangle with sides
of corresponding lengths. You could use a dynamic geometry package
to do this.

Do it again with three more lengths.

And again but instead of drawing the triangle put a question
mark. After some thinking time, encourage a member of the group to
come up and draw the triangle.

Finally, list three lengths that will not work followed by a
question mark and after time has been taken to realise the
impossibility, discuss why this is the case as a group.

Now pose the problem.

Working in small groups the challenge will be to employ
systematic approaches as well as applying the triangle
inequality.

Take opportunities to pull together different ideas for
recording, including the use of nets and working
systematically.

Key questions

How do you know you have tried all possibilities?

Is it possible to construct more than one tetrahedron?

Can you find six lengths which will give more than one
tetrahedron?

Possible extension

The problem Triangles
to Tetrahedra requires students to work systematically to
generate all possible tetrahedra from four particular
triangles.

Or there is the problem : Sliced .
This is a challenging next step in this kind of
visualisation.

Possible support

Use construction straws of equivalent lengths to make (or fail
to make) triangles and tetrahedra.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.